Let \(U_k \in ^{n \times k}\)  and \(V_k^\mathsf {T} \in ^{k \times d}\)  be rank \(k\)  matrices such that \(\Vert U_kV_k^\mathsf {T} - A\Vert _{1,2} = \textnormal {SubApx}_{k,1}(A)\) . [1]} show that if \(S\)  is a lopsided \(\epsilon \) -embedding for matrices \((V_k,A^\mathsf {T})\)  with respect to the norm \(\Vert \cdot \Vert _h\) , then \(\min _{\text{rank-}k\ X}\Vert AS^\mathsf {T}X - A\Vert _{1,2} \le (1 + O(\epsilon ))\textnormal {SubApx}_{k,1}(A)\) . We show that a suitably scaled Gaussian random matrix \(\)  with \(\tilde{O}(k)\)  rows is a lopsided \((1/4)\) -embedding for matrices \((V_k,A^\mathsf {T})\)  with probability \(\ge 9/10\) . Thus, we have that with probability \(\ge 9/10\) ,
\(\min _{\text{rank-}k\ X}\Vert A^\mathsf {T}X - A\Vert _{1,2} \le (3/2)\textnormal {SubApx}_{k,1}(A).\) 
